Local resolution of ideals subordinated to a foliation

Abstract

Let M be a complex- or real-analytic manifold, $$\theta $$ be a singular distribution and $$\mathcal {I}$$ a coherent ideal sheaf defined on M. We prove the existence of a local resolution of singularities of $$\mathcal {I}$$ that preserves the class of singularities of $$\theta $$ , under the hypothesis that the considered class of singularities is invariant by $$\theta $$ -admissible blowings-up. In particular, if $$\theta $$ is monomial, we prove the existence of a local resolution of singularities of $$\mathcal {I}$$ that preserves the monomiality of the singular distribution $$\theta $$ .